Inflation and acceleration of the universe by nonlinear magnetic monopole fields

Abstract

Despite impressive phenomenological success, cosmological models are incomplete without an understanding of what happened at the big bang singularity. Maxwell electrodynamics, considered as a source of the classical Einstein field equations, leads to the singular isotropic Friedmann solutions. In the context of Friedmann-Robertson-Walker (FRW) spacetime, we show that singular behavior does not occur for a class of nonlinear generalizations of the electromagnetic theory for strong fields. A new mathematical model is proposed for which the analytical nonsingular extension of FRW solutions is obtained by using the nonlinear magnetic monopole fields.

Eur. Phys. J. C (2017) 77:105DOI 10.1140/epjc/s10052-017-4673-4

Regular Article - Theoretical Physics

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Web End = Ination and acceleration of the universe by nonlinear magnetic monopole elds

A. vgnaDepartment of Physics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, Northern Cyprus

http://orcid.org/0000-0002-9889-342X

Web End = Received: 13 June 2016 / Accepted: 5 February 2017 / Published online: 16 February 2017 The Author(s) 2017. This article is published with open access at Springerlink.com

Abstract Despite impressive phenomenological success, cosmological models are incomplete without an understanding of what happened at the big bang singularity. Maxwell electrodynamics, considered as a source of the classical Einstein eld equations, leads to the singular isotropic Fried-mann solutions. In the context of FriedmannRobertson Walker (FRW) spacetime, we show that singular behavior does not occur for a class of nonlinear generalizations of the electromagnetic theory for strong elds. A new mathematical model is proposed for which the analytical nonsingular extension of FRW solutions is obtained by using the nonlinear magnetic monopole elds.

1 Introduction

Cosmology has experienced remarkable advances in recent decades as a consequence of tandem observations of type-Ia supernovae and the cosmic microwave background. These observations suggest that cosmological expansion is accelerating [1]. The last two decades have witnessed enormous progress in our understanding of the source of this accelerated expansion. Furthermore, standard cosmology assumes that at the beginning, there must have been an initial singularity a breakdown in the geometric structure of space and time from which spacetime suddenly started evolving [2]. The standard cosmological model, with the source of Maxwell electrodynamics based on the Friedmann RobertsonWalker (FRW) geometry, leads to a cosmological singularity at a nite time in the past. In order to solve this puzzle, researchers have proposed many different mechanisms in the literature, such as nonminimal couplings, a cosmological constant, nonlinear Lagrangians with quadratic terms in the curvature, scalar ination elds, modied gravity theories, quantum gravity effects, and nonlinear electro-dynamics without modication of general relativity [327].

a e-mail: mailto:[email protected]

Web End [email protected]

One possible solution is to explore the evolution while avoiding the cosmic initial singularity contained in a given nonlinear effect of electromagnetic theory [10,13,16]. 1934, the nonlinear electrodynamics Lagrangian known as the BornInfeld Lagrangian was published by the physicists Max Born and Leopold Infeld [28]. This Lagrangian has the amusing feature of turning into Maxwell theory for low electromagnetic elds; moreover, the nonlinear Lagrangian is invariant under the duality transformation.

To solve the initial singularity problem, the early stages of the universe are assumed to be dominated by the radiation of nonlinear modications of Maxwells equations, which include a large amount of electromagnetic and gravitational elds. This is true inasmuch as strong magnetic elds in the early universe can cause deviations from linear electro-dynamics to nonlinear electrodynamics [6,7]. By following recently published procedures [16,17], in this paper the nonlinear magnetic monopole (NMM) elds are used to show the source of the acceleration of the universe without an initial singularity.

In this paper, we investigate a cosmological model of the universe with NMM elds coupled to gravity. The structure of the paper is as follows: In Sect. 1, we briey introduce NMM elds and consider the universe to be lled by pure nonlinear magnetic elds. In Sect. 2, we show that the universe accelerates without an initial singularity until it reaches the critical value of the scale factor. In Sect. 3, we check the classical stability of the universe under the deceleration phase. In Sect. 4, we report our conclusions.

2 Nonlinear magnetic monopole elds and a nonsingular FRW universe

In cosmology, magnetic elds have become more important since the wealth of observations of magnetic elds in the universe [29,30]. Magnetic elds are ubiquitous in under-

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105 Page 2 of 6 Eur. Phys. J. C (2017) 77 :105

standing the mysteries of the universe. The action of General Relativity (GR) coupled with NMM elds is given by

S = [integraldisplay] d4xg [bracketleftBigg]

M2pl

The energy momentum tensor

T = K F gLN M M (7)

with

K =

LN M M F

F (8)

can be used to obtain the general form of the energy density and the pressure p by varying the action as follows:

= LN M M + E2

LN M M F

2 R + LE M + LN M M [bracketrightBigg] , (1)

where MPl is the reduced Planck mass, R is the Ricci scalar, and is the ne-tuning parameter of LE M Maxwell elds.

LN M M is the Lagrangian of the NMM elds. From a conceptual point of view, this action has the advantage that it does not invoke any unobserved entities such as scalar elds, higher dimensions, or brane worlds. Furthermore, we can ignore the Maxwell elds ( = 0), because they are weak compared to

the dominant NMM elds in the very early epochs and ination. However, in the literature there are many proposals of cosmological solutions based on the Maxwell elds plus corrections [11,12,16,17,2426]. Herein, our main aim is to use this method to show that it yields an accelerated expansion phase for the evolution of the universe in the NMM eld regime. The new ingredient we add is a modication of the electrodynamics, which has no Maxwell limit. The Einstein eld equation and the NMM eld equation are derived from the action

R

(9)

and

p = LN M M [parenleftbig]2B2

E2

[parenrightbig]

3

LN M M F

. (10)

Here, it is assumed that the curvature is much larger than the wavelength of the electromagnetic waves, because the electromagnetic elds are the stochastic background. The average of the EM elds that are sources in GR have been used to obtain the isotropic FRW spacetime [32]. For this reason, one uses the average values of the EM elds as follows:

E = B = 0, Ei Bj = 0,

Ei E j =

1

2 g R = 2T, (2)

where 1 = MPl, and

g LN M M

F

F

13 E2gi j, Bi Bj =

[parenrightbigg] = 0. (3)

Note that Maxwell invariant is F = F F = (B2

E2)/2 > 0, and F is the eld strength tensor. The magnetic eld two-form is F = P sin()2d d or F = P sin()2

where P is the magnetic monopole charge. Furthermore, it is noted that in the weak eld limit the NMM Lagrangian does not yield the linear Maxwell Lagrangian [31]. In this work, following a standard procedure, we consider the pure magnetic eld under the following NMM eld Lagrangian suggested in Ref. [31]:

13 B2gi j. (11)

Note that later we omit the averaging brackets for sim

plicity. The most interesting case of this method occurs only when the average of the magnetic eld is not zero [32]. The universe has a magnetic property that the magnetic eld is frozen in the cosmology where the charged primordial plasma screens the electric eld. It is, in the pure nonlinear magnetic monopole case, clear that E2 = 0. Then Eqs. (9)

and (10) reduce to the simple following form:

= LN M M (12)

and

p = LN M M

2B2

3

6

2 (4)

where and l are the positive constants. The constant parameter will be xed according to other parameters. The NMM eld Lagrangian is folded into the homogeneous and isotropic FRW spacetime

ds2 = dt2 + a(t)2(dx2 + dy2 + dz2) (5) or it can be written as follows:

ds2 = dt2 + a(t)2[dr2 + r2(d2 + sin()2d2)] (6) where a is a scale factor, to investigate the effects on the acceleration of the universe.

LN M M

=

3/4

l2

1 + [parenleftBig]

F

LN M M F

. (13)

Then the FRW metric given in Eq. (5) is used to obtain Fried-manns equation as follows:

3 a

a =

2

2 ( + 3p) , (14)

where . over the a denotes the derivatives with respect to the cosmic time. The most important condition for the accelerated universe is + 3p < 0. Here, the NMM eld is

used as the main source of gravity. Using Eqs. (9) and (10),

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Eur. Phys. J. C (2017) 77 :105 Page 3 of 6 105

it is found that

+ 3p = 2LN M M 2B2

LN M M F

. (15)

12

27/4

[parenleftBig]

B2

3/4 1[parenrightbigg]

l2

3 . (16)

Thus, the requirement + 3p < 0 for the accelerating uni

verse is satised at (( B2 )3/4 <

= 1 + 23/4 [parenleftBig]B2

3/4

127/4 ), where there is a strong magnetic monopole eld in the early stages of the universe to force it to accelerate. By using the conservation of the energy-momentum tensor,

T = 0, (17) for the FRW metric given in Eq. (5), it is found that

+ 3 aa ( + p) = 0. (18)

Replacing and p from Eqs. (12) and (13), and integrating, the evolution of the magnetic eld under the change of the scale factor is obtained as follows:

B(t) =

B0a(t)2 . (19)

Then, by using Eqs. (12) and (13), the energy density and the pressure p can be written in the form of

=

6l2( )2 , (20)

p =

Fig. 1 Plot of the energy density and the pressure p versus (for the cases of a = 1, 0, )

R = 2( 3p). (25) The Ricci tensor squared R R and the Kretschmann scalar R R are also obtained:

R R = 4(2 + 3p2), (26)

R R = 4 [parenleftbigg]

1

4

6l2 ( )2 +

12 a423/4

l2B2 ( )3

a4B2, (21)

where

= 1 + 23/4 [parenleftbigg]

a4 B2

3/4. (22)

Note that from Eqs. (20) and (21), we obtain the energy density and the pressure p, but there is no singularity point at a(t) 0 and a(t) . Hence, one nds that, as shown

in Fig. 1,

lim

a(t)0

(t) =

532 + 2p + 3p2[parenrightbigg] . (27)

We study the Ricci scalar depending on the scale factor from Eq. (19) and take the limit of Eq. (25) to show that the nonsingular curvature, the Ricci tensor, and the Kretschmann scalar when the universe accelerates at a(t) 0 and at a(t) .

lim

a(t)0

R(t) =

242l2 , (28)

lim

a(t)0

R R =

1444l4 , (29)

lim

a(t)0

6l2 , lim

a(t)0

p(t) =

6l2 , (23)

lim

R R =

964l4 , (30)

lim

a(t)

p(t) = 0. (24)

From Eqs. (23) and (24), it is concluded that the energy density is equal to the negative of the pressure p ( = p) at

the beginning of the universe (a = 0), similarly to a model

of the CDM. The absence of singularities is also shown in the literature [16,17] by using a different model of nonlinear electrodynamics.

The Ricci scalar, which represents the curvature of space-time, is calculated by using Einsteins eld equation (2) and the energy-momentum tensor,

a(t)

(t) = lim

a(t)

R(t) = lim

a(t)

R R = lim

a(t)

R R = 0.

(31)

In the future, the acceleration of the universe will stop at an innite time, and then the spacetime will become at, without any singularities. The critical scale factor to show the boundary of the universe acceleration is obtained using

Eqs. (15) and (19) as follows: a(t) < ac =

1

27/12

. Hence, the acceleration of the universe is zero at the critical value of

B0

4

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105 Page 4 of 6 Eur. Phys. J. C (2017) 77 :105

the scale factor a = ac, and at the early time we show that

the universe accelerates until the critical scale factor a = ac,

which is suggested by the model describing ination without singularities.

3 Evolution of the universe

In this section, we study the dynamics of the universe by using Einsteins equations and the energy density given in Eq. (21).We are interested in the early regime of the universe without using the dustlike matter. First, with the help of the second Friedmann equation, we nd the evolution of the scale factor as given by

[parenleftbigg] a

a

a2 , (32)

where = 0, +1 and 1 depending on the geometry of the

universe (at, closed, and open, respectively). Then, using the Eq. (14) the value of the critical scale factor is obtained as ac =

1

27/12

=

2 2

3

B0

4

. Now, one shows that a cosmic time [33] is calculated by

t =

1 3A

23/4C3/4a3 + 3 ln (a)[parenrightBig] +t0, (33)

where t0 is a constant of integration which gives only the shift in time, A = 22l2 and C =

B20 =

1 a4

c . Note the assumptions that the universe is at ( = 0) and the integral constant is

t0 = 0. Furthermore, Eq. (33) can be written in the units of

the critical scale factor as

t =

1 3A

Fig. 2 Plot of the scale factor a versus the time t (for A = 1 and

C = 1)

noted that small cosmic time in the regime of the early universe depends on the NMM elds and they play essential role for the evolution of the universe in the early regime. This shows also that as a0 0.77ac < ac, the universe experi

ences accelerating expansion, without need for dark energy models. The acceleration of the universe begins at the initial radius of the universe a0 and it stops at the critical value ac, where the acceleration of the universe is zero a = 0. After

the acceleration stops, the universe decelerates until the big crunch.

Furthermore, for the positive time cases there is no singularities for the spatial curvature K =

23/4 a3a3c +

3 ln(a)

, (34)

so one nds

a(t) = exp [bracketleftbigg]

13 LambertW [parenleftbigg]

[parenrightbigg] + At[bracketrightbigg] , (35)

and we consider t = 0 to nd the equation for the radius of

the universe,

1 3A

23/4 a3a3c +

23/4e3A t

a3c

1a2 when t 0and

t ; however, when t , clearly the scale of spatial

curvature goes to innity leaving the closed universe,

lim

t0

K = 1.67, lim

t

K = 0, lim

t

K = . (38)

3.1 A test of causality with speed of the sound

A well-known way to test the causality of the universe in order to survive, is by using the speed of the sound, which must be less than the local light speed, cs 1 [34]. The next

requirement is based on c2s > 0, positive value of the square sound speed. Those requirements satisfy a classical stability requirement. The square of the sound speed is obtained from Eqs. (9) and (10):

c2s =

d p d =

[parenrightbigg] = 0, (36)

where the solution is found to be

a0 = a(t = 0) = exp [bracketleftBigg]

3 ln (a)

13 LambertW [parenleftBigg]

3/425/2

B3/20

[parenrightBigg][bracketrightBigg] .

(37)

The function of a0 is a radius of the universe. Hence, the Eq. (37) represents the phase of the universe without any singularity (t = 0). The cosmic time calculated in Eq. (34)

has no singularity, and the scale factor as a function of time has almost exponential behavior as shown in Fig. 2. It is

d p/dF

d/dF

(39)

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Eur. Phys. J. C (2017) 77 :105 Page 5 of 6 105

2 [parenleftbigg]5 B2023/4 4[radicalbigg]

a4

B20 + 2a4[parenrightbigg]

1

4

a4 B2

=

4

0 , (40)

3B20 [parenleftBigg]1 + 23/4 [parenleftbigg]

a4 B20

3/4

[parenrightBigg]

and the classical stability (c2s > 0) occurs at

5 B2023/4 4[radicalBigg]

a4

B20 +

2a4 < 0. (41)

The inequality of cs 1 is satised in Eq. (39) for a posi

tive values of ( > 0). It should be noted that the magnetic eld strength B0 can have any value. Then, to satisfy the classical stability, in Eq. (41) we nd the limit of the scale factor as a(t) > 51/321/12 B0

4

. So the deceleration of the universe occurs at this stage after reach the critical value of acceleration nished at ac =

1

27/12

1.81

B0

4

B0

4

0.67

B0

4

. Therefore, a superluminal uctuation of the universe (cs 1) does

only occurs at the early deceleration phase of the universe in the cosmological model with NMM elds. Furthermore, this model has a classical instability at a(t) < 1.81B0

4

. This

instability can be explained by the ination period and the short universe deceleration time, a result of the uncontrollable growth of the energy density perturbation.

4 Conclusion

In this paper, we used the model of NMM elds with parameters and l for the sources of the gravitational eld. This model is not scale-invariant because of the free parameters and l, so the energy-momentum tensor is not zero. We consider the universe to be magnetic and to accelerate with the help of NMM eld sources. After the ination period, it was shown that the universe is homogeneous and isotropic. The acceleration of the universe is bounded at a(t) < ac(t) =

1

27/12

B0

4

.

We also showed that, at the time of the Big Bang, there was no singularity in the energy density, pressure, or curvature terms. After some time, the universe approaches at spacetime. We checked causality and found that it satises the classical stability where the speed of sound should be less than the local light speed. Hence, nonlinear sources, such as NMM elds at the early regime of the universe, allow accelerated expansion with ination and without dark energy. This model of NMM elds can also be used to describe the evolution of the universe. We noted that at the weak NMM eld, there is no Maxwells limit, so the ination and the acceleration of the universe can be analyzed by using different types of elds. We manage to smooth the singularity of the magnetic universe by using only the NMM elds that were strong

in the accelerated phase of the universe. In our model, in the early regime of the universe, NMM elds are very strong, making the effects of the usual electromagnetic elds negligible. We leave for a future publication the use of NMM elds with the usual Maxwell elds, and scalar elds, to investigate this problem more deeply. Another future project is to nd the relationship between the different types of NMM elds and the possible existence of wormholes in the universe and their effect of Hawking radiation in relation to our previous work [3538].

Acknowledgements The author would like to thank Prof. Dr. Mustafa Halilsoy for reading the manuscript and giving valuable suggestions. The author is grateful to the editor and anonymous referees for their valuable and constructive suggestions.

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